How to run blender video editor on a weak graphics card?
Not having your exact hardware, it is really difficult to give an exact answer, or even good suggestions. You didn't even describe your pc specs or your card model, drivers version. It could help.

But I think you could, as a last resort, try a software opengl emulation. It won't be performant, it won't be fast and it could have (other) glitches, but at least you could try. 

How? In simple terms, usually blender relies on the (usually much powerful) opengl hardware implementation but there are software implementation that are run by your cpu, and they could support better opengl levels, hopefully better than your gpu or driver.

The most used, I think is Mesa (https://mesa3d.org/) and you should build (compile) it to obtain a dll. But you can also find some .dll pre-built around... just be careful about the source you're relying on!

Here is a developer's thread that explains roughly how to do this (note the link suggested is now dead!):

https://developer.blender.org/T43077


  Download opengl32.dll from
  http://download.blender.org/ftp/incoming/softwaregl/ and put it into
  the same folder as the blender.exe. This will switch blender to fully
  software opengl mode, avoiding possible issues with
  driver/configuration


I found a steam thread that has working links, apparently


  for 32bit
  http://download.blender.org/ftp/sergey/softwaregl/win32/opengl32.dll
  and for 64
  http://download.blender.org/ftp/sergey/softwaregl/win64/opengl32.dll
  once downloaded just put in the directory of blender


https://steamcommunity.com/app/365670/discussions/0/598198356178829559/

See this thread also 

https://www.opengl.org/discussion_boards/showthread.php/138029-OpenGL32-dll

I used this approach once (on linux, on thinkpad R60 with a crappy intel card) and it kinda worked, although the most complex functions in blender had issues, but I used this way until I had a new pc, with a much better card, to use the (at the time) newest blender versions... So, you have to try, and maybe report here your findings, to help others! 

Good luck!